Answer:
7 feet and 24 feet
Step-by-step explanation:
In the right triangle, the hypotenuse is 25 feet long. The area of this triangle is 84 square feet.
Let x feet and y feet be the lengths of triangle's legs.
By the Pythagorean theorem,
[tex]x^2+y^2=25^2\\ \\x^2+y^2=625[/tex]
The area of the right triangle is half the product of its legs, thus
[tex]84=\dfrac{1}{2}xy\\ \\xy=168[/tex]
Solve the system of two equations:
[tex]\left\{\begin{array}{l}x^2+y^2=625\\ \\xy=168\end{array}\right.[/tex]
From the second equation:
[tex]x=\dfrac{168}{y}[/tex]
Substitute it into the first equation:
[tex]\left(\dfrac{168}{y}\right)^2+y^2=625\\ \\168^2+y^4=625y^2\\ \\y^4-625y^2+168^2=0\\ \\D=(-625)^2-4\cdot 168^2=(625-2\cdot 168)(625+2\cdot 168)=289\cdot 961=17^2\cdot 31\\ \\\sqrt{D}=17\cdot 31=527\\ \\y^2_{1,2}=\dfrac{-(-625)\pm 527}{2}=49,\ 576\\ \\y_1=7,\ y_2=-7,\ y_3=24,\ y_4=-24[/tex]
The length of the leg cannot be negative, so
[tex]y_1=7\Rightarrow x_1=\dfrac{168}{7}=24\\ \\y_2=24\Rightarrow x_2=\dfrac{168}{24}=7[/tex]
